In online-setting, we usually allow exponential time to think and come up with a strategy. On the other hand, in offline-setting, we might care about solving a particular problem optimally, or as good as possible, in polynomial time.
Therefore, there is a chance that for some problems, we can actually come up with a strategy that gives us a competitive ratio that is better than best offline-approximation.
These problems could be crucial to study to obtain some insight understanding on online algorithms. Please let me know if you know of some examples.